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March 9, 2014 / porton

“Funcoidal” reloids, a new research idea

Just today I’ve got the idea of the below conjecture:

Definition I call funcoidal such reloid \nu that

\mathcal{X} \times^{\mathsf{RLD}} \mathcal{Y} \not\asymp \nu \Rightarrow \\ \exists \mathcal{X}' \in \mathfrak{F}^{\mathrm{Base} ( \mathcal{X})} \setminus \{ 0 \}, \mathcal{Y}' \in \mathfrak{F}^{\mathrm{Base} ( \mathcal{Y})} \setminus \{ 0 \} : ( \mathcal{X}' \sqsubseteq \mathcal{X} \wedge \mathcal{Y}' \sqsubseteq \mathcal{Y} \wedge \mathcal{X}' \times^{\mathsf{RLD}} \mathcal{Y}' \sqsubseteq \nu)
for every \mathcal{X} \in \mathfrak{F}^{\mathrm{Src}\, \nu}, \mathcal{Y} \in \mathfrak{F}^{\mathrm{Dst}\, \nu}.

Easy to prove proposition:

Proposition A reloid \nu is funcoidal iff x \times^{\mathsf{RLD}} y \not\asymp \nu \Rightarrow x \times^{\mathsf{RLD}} y \sqsubseteq \nu for every ultrafilters x and y on respective sets.

Conjecture ( \mathsf{RLD})_{\mathrm{in}} is a bijection from \mathsf{FCD}( A ; B) to the set of funcoidal reloids from A to B.


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